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\title{Mathematical Writing Exercise Chapter 3.8 - 3.14}
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%\date{2024 年 4 月 9 日}
\date{April 9, 2024}

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\begin{enumerate}

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\item  %Problem 01 ch3.8.
True or False. (Ambiguous `This' and `It')
\begin{enumerate}[label={(\arabic*)}]

\item  {\color{red}A requirement of good writing is to make clear to the reader, at all times, what is the entity under discussion. }
 \dotfill (\,\,\,\,\,\,\,\,\,\,)
 
\item  {\color{red}\it This} phrases such as ``This is a consequence of Theorem 2'' should be used with caution as they can force the reader to backtrack to find what ``this'' refers to. 
Often it helps to insert an appropriate noun after {\color{red}\it this}. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)
 
\item  {\color{red}\it It} can also be ambiguous: in the sentence ``Condition 3 is not satisfied for the steepest descent method, which is why we do not consider it further'' we cannot tell whether it is the condition or the method that is not being pursued. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)
 
\item  Other pronouns, such as ``these'', ``those'', and ``they'', also need to be used carefully to avoid ambiguity.
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 02 ch3.9
True or False. (Capitalization)
\begin{enumerate}[label={(\arabic*)}]

\item  {\color{red}Words that are derived from a person's name inherit the capitalization. }
Thus: Gaussian elimination, Hamiltonian system, Hermitian matrix, Jacobian matrix, Lagrangian function, Euler's method, and so on. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The incorrect form `hermitian' is sometimes seen. 
The Lax Equivalence Theorem is quite different from a lax equivalence theorem! 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Some (but not all) dictionaries list `abelian' with a small `a', showing that eponymous adjectives can gradually become accepted in uncapitalized form. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  There does not seem to be a standard rule for when to capitalize the word following a colon. 
Theodore Bernstein [34] and Donald Knuth [205, p. 11] both suggest the useful convention of capitalizing when the phrase following the colon is a full sentence.
 \dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 03 ch3.10
True or False. (Common Misspellings or Confusions)
\begin{enumerate}[label={(\arabic*)}]

\item  The errors shown in the table seem to be common in mathematical writing. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)
 
\item  The misspellings marked with an asterisk are genuine words and are either the plural of or have a different meaning from the corresponding word in the left column.
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  {\color{red}\it Supersede} is the only English word ending in -sede.
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Some, but not all, dictionaries give the -oes ending as an alternative spelling for the plural noun of the word `zero'. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  One web page I visited describes `seperate' as the most common misspelling on the internet and lose/loose as the second most common. 
In a Google search I found one occurrence of `seperate' for every 24 occurrences of `separate'. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The journal {\color{red}\it Physical Review Letters} started spelling {\color{red}\it Lagrangian} as {\color{red}\it Lagrangean} in mid 1985, a change which is incorrect according to most dictionaries. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  David Mermin, a Cornell physicist, spotted the switch and wrote an article criticizing it [238]. 
The journal reverted to Lagrangian. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  According to Douglas Mcllroy [234], on most days at Bell Laboratories someone misspelled the word {\color{red}\it accommodate}, in one of seven incorrect ways.
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\newpage

\begin{table}[ht!]
\centering
\caption{Common errors. Asterisk denotes a genuine word.\vspace{0.3cm}}
\begin{tabular}{ll}\hline 
Correct / intended & Misspelling \\ \hline \hline 
analogous & analagous \\ \hline 
criterion & criteria* (plural of criterion) \\ \hline 
dependent & dependant* \\ \hline 
discrete & discreet* \\ \hline 
Probenius & Probenious \\ \hline 
idiosyncrasy & idiosyncracy \\ \hline 
in practice & in practise \\ \hline 
led (past tense of lead) & lead* \\ \hline 
lose & loose* (very common) \\ \hline 
phenomenon & phenomena* (plural of phenomenon) \\ \hline 
preceding & preceeding \\ \hline 
principle & principal* \\ \hline 
propagation & propogation \\ \hline 
referring & refering \\ \hline 
Riccati & Ricatti \\ \hline 
separate & seperate \\ \hline 
supersede & supercede \\ \hline 
zeros & zeroes \\ \hline 
\end{tabular}
\end{table}

\vspace{0.2cm}

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\item  %Problem 04 ch3.11. 
True or False. (Consistency)
\begin{enumerate}[label={(\arabic*)}]

\item  {\color{red}It is important to be consistent. }
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Errors of consistency often go unnoticed, but they can be puzzling to the reader. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Don't refer to ker(A) as the null-space or null(A) as the kernel - stick to matching synonyms. .......
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  If you use the term `Cholesky factorization', don't refer to the `Cholesky decomposition' in the same work. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  If words have alternative spellings, stick to one: don't use both orthogonalise and orthogonalize, and if you use orthogonalize also use, for example, optimize, not optimise. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  But note that not all -ise words can be spelt with -ize; examples are listed in 3.34.
 \dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

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\item  %Problem 05 ch3.12. 
True or False. (Contractions)
\begin{enumerate}[label={(\arabic*)}]

\item  Contractions such as {\color{red}\it it's}, {\color{red}\it let's}, {\color{red}\it can't}, and {\color{red}\it don't} are not used in formal works such as papers and theses, but they are acceptable in popular articles if used sparingly. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Note the distinction between the contraction {\color{red}\it it's} (short for {\color{red}\it it is}) and the possessive {\color{red}\it its}: ``It's raining'', ``A matrix is singular if its determinant is zero''. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  One editor comments that the two most frequent errors she encounters are the use of {\color{red}\it it's} for {\color{red}\it its} and incorrect punctuation surrounding {\color{red}\it however} [341, p. 39].
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

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\item  %Problem 06  ch3.13. 
True or False. (Dangling Participles)
\begin{enumerate}[label={(\arabic*)}]

\item  What is wrong with the following sentence?
\begin{center}\fbox{
\begin{minipage}{12cm}
Substituting (3) into (7), the integral becomes $\pi^2/9$.
\end{minipage}
} \end{center}
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
The sentence suggests that the integral makes the substitution. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The error is that the intended subject (``we'') of the participle {\color{red}\it substituting} is not present in the sentence. 
Rewritten and unambiguous versions are
\begin{center}\fbox{
\begin{minipage}{12cm}
Substituting (3) into (7), we find that the integral is $\pi^2/9$.

When (3) is substituted into (7), the integral becomes $\pi^2/9$.
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A similar example that is less obviously wrong is
\begin{center}\fbox{
\begin{minipage}{12cm}
When deriving parallel algorithms, the model of computation must be considered carefully. 
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  {\color{red}Dangling participles} are usually not ambiguous when read in context, but they can be distracting: 
\begin{center}\fbox{
\begin{minipage}{12cm}
A bug was found in the program using random test data. 
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Here is another example of a different type.
\begin{center}\fbox{
\begin{minipage}{12cm}
Being stiff, the Runge-Kutta routine required a large amount of CPU time to solve the differential equation. 
\end{minipage}
} \end{center}
Here, the problem is that the sentence incorrectly implies that the Runge-Kutta routine, not the differential equation, is stiff. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  There are several ways to rewrite the sentence. One that preserves the emphasis is 
\begin{center}\fbox{
\begin{minipage}{12cm}
Because the differential equation is stiff, the Runge-Kutta routine required a large amount of CPU time to solve it. 
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  {\color{red} Certain participial constructions} are idiomatic and hence are regarded as acceptable: 
\begin{center}\fbox{
\begin{minipage}{12cm}
Assuming $G(x^*)$ is positive definite, $x^*$ is a minimum point for $F$.

Considering the large dimension of the problem, convergence was obtained in remarkably few iterations. 

Strictly speaking, the bound holds only for $n\epsilon < 1$.
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

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\item  %Problem 07 ch3.14.1. 
True or False. (Distinctions: Affect, Effect. )
\begin{enumerate}[label={(\arabic*)}]

%\item  Affect, Effect. 

\item  {\color{red}\it Affect} is a verb meaning to produce a change. {\color{red}\it Effect} is a noun meaning the result of a change. Examples: 
\begin{center}\fbox{
\begin{minipage}{12cm}
Multiple roots affect the convergence rate of Newton's method. 

One effect of multiple roots is to reduce the convergence rate of Newton's method to linear. 
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  {\color{red}\it Effect} is also a verb meaning to bring about (as in ``to effect a change''), but it is rarely needed in this form in mathematical writing.
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

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\item  %Problem 08 ch3.14.2. 
True or False. (Distinctions: Alternate, Alternative. )
\begin{enumerate}[label={(\arabic*)}]

%\item  Alternate, Alternative. 

\item  An {\color{red}\it alternative} is one of several options. In American English, but not British English, the noun {\color{red}\it alternate} has the same meaning. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  As a verb, {\color{red}\it alternate} means to change repeatedly from one thing to another. 
Compare ``While writing his thesis the student alternated between elation and misery'' with ``The first attempt to prove the theorem failed, so an alternative method of proof was tried''. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

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\item  %Problem 09 ch3.14.3. 
True or False. (Distinctions: Compare with, Compare to. )
\begin{enumerate}[label={(\arabic*)}]

%\item  Compare with, Compare to. 

\item  {\color{red}\it Compare with} analyzes similarities and differences between two things, whereas {\color{red}\it compare to} states a resemblance between them. Examples: 
\begin{center}\fbox{
\begin{minipage}{12cm}
We now {\color{red}\it compare} Method A {\color{red}\it with} Method B. 

Shakespeare {\color{red}\it compared} the world {\color{red}\it to} a stage. 

Shall I {\color{red}\it compare} thee {\color{red}\it to} a summer's day?
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  As Bill Bryson [53] explains, ``Unless you are writing poetry or love letters, {\color{red}\it compare with} is usually the expression you want.'' 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  {\color{red}\it Compare and} is an alternative to {\color{red}\it compare with}: ``We now {\color{red}\it compare} Method A {\color{red}\it and} Method B'' or, better, ``We now {\color{red}\it compare} Methods A {\color{red}\it and} B.''
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

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\item  %Problem 10 ch3.14.4. 
True or False. (Distinctions: Compose, Comprise, Constitute. )
\begin{enumerate}[label={(\arabic*)}]

%\item  Compose, Comprise, Constitute. 

\item  {\color{red}\it Compose} means to make up, {\color{red}\it comprise} means to consist of. ``Comprised of'' is always incorrect. ......
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Thus, ``the course {\color{red}\it is composed of} three topics'' or ``the course {\color{red}\it comprises} three topics,'' but not ``the course is comprised of three topics.'' 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  {\color{red}\it Constitute} is a transitive verb used in the reverse sense: ``these three topics {\color{red}\it constitute} the course.'' 
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 11 ch3.14.5. 
True or False. (Distinctions: Due to, Owing to, Because of. )
\begin{enumerate}[label={(\arabic*)}]

%\item  Due to, Owing to, Because of. 

\item  {\color{red}\it Due to} is best reserved for uses such as ``The instability {\color{red}\it is due to} a rank deficient submatrix,'' where {\color{red}\it due} functions as an adjective and modifies a noun (``the instability''). 
If no such noun can be identified use ``{\color{red}\it because of}''. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A test for correct usage is that it should be possible to replace ``{\color{red}\it due to}'' by ``{\color{red}\it attributable to}'' or ``{\color{red}\it caused by}''. Here is an example.
\begin{center}\fbox{
\begin{minipage}{12cm}
Incorrect: Pivoting is necessary due to the lack of definiteness of the matrix. 

Correct: Pivoting is necessary because the matrix is not  definite.
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  To start a sentence use ``{\color{red}\it because of}'' or ``{\color{red}\it owing to}'' instead of {\color{red}\it due to}: 
``{\color{red}\it Owing to} a rank deficient submatrix the computed result was inaccurate.''
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 12 ch3.14.6.  
True or False. (Distinctions: Fewer, Less. )
\begin{enumerate}[label={(\arabic*)}]

%\item  Fewer, Less. 

\item  {\color{red}\it Less} refers to quantity, amount, or size, {\color{red}\it fewer} to number. Thus ``the zeros of $f(x)$ are {\color{red}\it less than} those of $g(x)$'' means that if $a$ is a zero of $f$ and $b$ a zero of $g$ then $a < b$, whereas ``the zeros of $f(x)$ are {\color{red}\it fewer than} those of $g(x)$'' means that $g$ has more zeros than $f$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Bryson [53] states the rule of thumb that {\color{red}\it less} should be used with singular nouns and {\color{red}\it fewer} with plural nouns: less research, less computation, fewer graduates, fewer papers.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

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\end{enumerate}


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